Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x + \sin y\right) + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \sin y\right) + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y, z, x + \sin y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (cos y) z (+ x (sin y))))

C

double code(double x, double y, double z) {
	return fma(cos(y), z, (x + sin(y)));
}

Julia

function code(x, y, z)
	return fma(cos(y), z, Float64(x + sin(y)))
end

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot \cos y + \left(x + \sin y\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \cos y} + \left(x + \sin y\right) \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\cos y \cdot z} + \left(x + \sin y\right) \]

   5. lower-fma.f64 99.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x + \sin y\right)} \]

   6. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x + \sin y}\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

   8. lower-+.f64 99.9

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y + x\right)} \]

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\cos y, z, x + \sin y\right) \]
6. Add Preprocessing

Alternative 2: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+125}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 1020000000:\\ \;\;\;\;\mathsf{fma}\left(1, z, \mathsf{fma}\left(\frac{\sin y}{x}, x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e+125)
   (* z (cos y))
   (if (<= z 1020000000.0)
     (fma 1.0 z (fma (/ (sin y) x) x x))
     (fma (cos y) z (+ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e+125) {
		tmp = z * cos(y);
	} else if (z <= 1020000000.0) {
		tmp = fma(1.0, z, fma((sin(y) / x), x, x));
	} else {
		tmp = fma(cos(y), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e+125)
		tmp = Float64(z * cos(y));
	elseif (z <= 1020000000.0)
		tmp = fma(1.0, z, fma(Float64(sin(y) / x), x, x));
	else
		tmp = fma(cos(y), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e+125], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1020000000.0], N[(1.0 * z + N[(N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.0000000000000001e125

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 92.1

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -9.0000000000000001e125 < z < 1.02e9

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \cos y + \left(x + \sin y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \cos y} + \left(x + \sin y\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} + \left(x + \sin y\right) \]

      5. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x + \sin y\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x + \sin y}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

      8. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y + x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x \cdot \left(1 + \frac{\sin y}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, x \cdot \color{blue}{\left(\frac{\sin y}{x} + 1\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\frac{\sin y}{x} \cdot x + 1 \cdot x}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \frac{\sin y}{x} \cdot x + \color{blue}{x}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\mathsf{fma}\left(\frac{\sin y}{x}, x, x\right)}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \mathsf{fma}\left(\color{blue}{\frac{\sin y}{x}}, x, x\right)\right) \]

      6. lower-sin.f64 99.9

         \[\leadsto \mathsf{fma}\left(\cos y, z, \mathsf{fma}\left(\frac{\color{blue}{\sin y}}{x}, x, x\right)\right) \]

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\mathsf{fma}\left(\frac{\sin y}{x}, x, x\right)}\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \mathsf{fma}\left(\frac{\sin y}{x}, x, x\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 97.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \mathsf{fma}\left(\frac{\sin y}{x}, x, x\right)\right) \]

   if 1.02e9 < z

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \cos y + \left(x + \sin y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \cos y} + \left(x + \sin y\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} + \left(x + \sin y\right) \]

      5. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x + \sin y\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x + \sin y}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

      8. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y + x\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x + y}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]

      2. lower-+.f64 87.1

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]

   7. Applied rewrites 87.1%

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+125}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 1020000000:\\ \;\;\;\;\mathsf{fma}\left(1, z, \mathsf{fma}\left(\frac{\sin y}{x}, x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+125}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-72}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-59}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e+125)
   (* z (cos y))
   (if (<= z -1.12e-72)
     (+ x z)
     (if (<= z 8e-59) (+ x (sin y)) (fma (cos y) z (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e+125) {
		tmp = z * cos(y);
	} else if (z <= -1.12e-72) {
		tmp = x + z;
	} else if (z <= 8e-59) {
		tmp = x + sin(y);
	} else {
		tmp = fma(cos(y), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e+125)
		tmp = Float64(z * cos(y));
	elseif (z <= -1.12e-72)
		tmp = Float64(x + z);
	elseif (z <= 8e-59)
		tmp = Float64(x + sin(y));
	else
		tmp = fma(cos(y), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e+125], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.12e-72], N[(x + z), $MachinePrecision], If[LessEqual[z, 8e-59], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.0000000000000001e125

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 92.1

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -9.0000000000000001e125 < z < -1.12000000000000005e-72

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z + x} \]

      2. lower-+.f64 83.8

         \[\leadsto \color{blue}{z + x} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{z + x} \]

   if -1.12000000000000005e-72 < z < 8.0000000000000002e-59

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\sin y + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sin y + x} \]

      3. lower-sin.f64 97.6

         \[\leadsto \color{blue}{\sin y} + x \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\sin y + x} \]

   if 8.0000000000000002e-59 < z

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \cos y + \left(x + \sin y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \cos y} + \left(x + \sin y\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} + \left(x + \sin y\right) \]

      5. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x + \sin y\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x + \sin y}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

      8. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y + x}\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y + x\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{x + y}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]

      2. lower-+.f64 86.5

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]

   7. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+125}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-72}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-59}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -9 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-72}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 11200000000:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -9e+125)
     t_0
     (if (<= z -1.12e-72)
       (+ x z)
       (if (<= z 11200000000.0) (+ x (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -9e+125) {
		tmp = t_0;
	} else if (z <= -1.12e-72) {
		tmp = x + z;
	} else if (z <= 11200000000.0) {
		tmp = x + sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * cos(y)
    if (z <= (-9d+125)) then
        tmp = t_0
    else if (z <= (-1.12d-72)) then
        tmp = x + z
    else if (z <= 11200000000.0d0) then
        tmp = x + sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -9e+125) {
		tmp = t_0;
	} else if (z <= -1.12e-72) {
		tmp = x + z;
	} else if (z <= 11200000000.0) {
		tmp = x + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -9e+125:
		tmp = t_0
	elif z <= -1.12e-72:
		tmp = x + z
	elif z <= 11200000000.0:
		tmp = x + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -9e+125)
		tmp = t_0;
	elseif (z <= -1.12e-72)
		tmp = Float64(x + z);
	elseif (z <= 11200000000.0)
		tmp = Float64(x + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -9e+125)
		tmp = t_0;
	elseif (z <= -1.12e-72)
		tmp = x + z;
	elseif (z <= 11200000000.0)
		tmp = x + sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9e+125], t$95$0, If[LessEqual[z, -1.12e-72], N[(x + z), $MachinePrecision], If[LessEqual[z, 11200000000.0], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.0000000000000001e125 or 1.12e10 < z

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 85.9

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -9.0000000000000001e125 < z < -1.12000000000000005e-72

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z + x} \]

      2. lower-+.f64 83.8

         \[\leadsto \color{blue}{z + x} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{z + x} \]

   if -1.12000000000000005e-72 < z < 1.12e10

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\sin y + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sin y + x} \]

      3. lower-sin.f64 95.2

         \[\leadsto \color{blue}{\sin y} + x \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\sin y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+125}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-72}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 11200000000:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.0037:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) \cdot z + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.0037)
     t_0
     (if (<= y 5e-14)
       (+
        (* (fma (fma 0.041666666666666664 (* y y) -0.5) (* y y) 1.0) z)
        (+ x y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.0037) {
		tmp = t_0;
	} else if (y <= 5e-14) {
		tmp = (fma(fma(0.041666666666666664, (y * y), -0.5), (y * y), 1.0) * z) + (x + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.0037)
		tmp = t_0;
	elseif (y <= 5e-14)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(y * y), -0.5), Float64(y * y), 1.0) * z) + Float64(x + y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0037], t$95$0, If[LessEqual[y, 5e-14], N[(N[(N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0037000000000000002 or 5.0000000000000002e-14 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\sin y + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sin y + x} \]

      3. lower-sin.f64 63.3

         \[\leadsto \color{blue}{\sin y} + x \]

   5. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\sin y + x} \]

   if -0.0037000000000000002 < y < 5.0000000000000002e-14

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + y\right)} + z \cdot \cos y \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]

      2. lower-+.f64 100.0

         \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(y + x\right) + z \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right) \]

      10. lower-*.f64 100.0

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0037:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) \cdot z + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1050.0)
   (+ x z)
   (if (<= y 1.7e+35)
     (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1050.0) {
		tmp = x + z;
	} else if (y <= 1.7e+35) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1050.0)
		tmp = Float64(x + z);
	elseif (y <= 1.7e+35)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1050.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 1.7e+35], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1050 or 1.7000000000000001e35 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z + x} \]

      2. lower-+.f64 40.1

         \[\leadsto \color{blue}{z + x} \]

   5. Applied rewrites 40.1%

      \[\leadsto \color{blue}{z + x} \]

   if -1050 < y < 1.7000000000000001e35

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z\right) + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + \left(x + z\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y} + \left(x + z\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, x + z\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1}, y, x + z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y} + 1, y, x + z\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right)}, y, x + z\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z}, y, 1\right), y, x + z\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right)}, y, 1\right), y, x + z\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \color{blue}{\frac{-1}{2} \cdot z}\right), y, 1\right), y, x + z\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right), y, 1\right), y, \color{blue}{z + x}\right) \]

      12. lower-+.f64 95.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, \color{blue}{z + x}\right) \]

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1050.0)
   (+ x z)
   (if (<= y 1.32e+20) (fma (fma (* -0.5 y) z 1.0) y (+ x z)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1050.0) {
		tmp = x + z;
	} else if (y <= 1.32e+20) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1050.0)
		tmp = Float64(x + z);
	elseif (y <= 1.32e+20)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1050.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 1.32e+20], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1050 or 1.32e20 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z + x} \]

      2. lower-+.f64 39.9

         \[\leadsto \color{blue}{z + x} \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{z + x} \]

   if -1050 < y < 1.32e20

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + y \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z\right) + y \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) + \left(x + z\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot y} + \left(x + z\right) \]

      4. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{2} \cdot \color{blue}{\left(z \cdot y\right)}\right) \cdot y + \left(x + z\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \color{blue}{\left(\frac{-1}{2} \cdot z\right) \cdot y}\right) \cdot y + \left(x + z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \left(\frac{-1}{2} \cdot z\right) \cdot y, y, x + z\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot z\right) \cdot y + 1}, y, x + z\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \left(z \cdot y\right)} + 1, y, x + z\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(y \cdot z\right)} + 1, y, x + z\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot y\right) \cdot z} + 1, y, x + z\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot y, z, 1\right)}, y, x + z\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot y}, z, 1\right), y, x + z\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z, 1\right), y, \color{blue}{z + x}\right) \]

      14. lower-+.f64 96.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, \color{blue}{z + x}\right) \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15600000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -15600000.0)
   (+ x z)
   (if (<= y 1.7e+35)
     (fma (fma (* -0.16666666666666666 y) y 1.0) y (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -15600000.0) {
		tmp = x + z;
	} else if (y <= 1.7e+35) {
		tmp = fma(fma((-0.16666666666666666 * y), y, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -15600000.0)
		tmp = Float64(x + z);
	elseif (y <= 1.7e+35)
		tmp = fma(fma(Float64(-0.16666666666666666 * y), y, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -15600000.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 1.7e+35], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.56e7 or 1.7000000000000001e35 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z + x} \]

      2. lower-+.f64 40.3

         \[\leadsto \color{blue}{z + x} \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{z + x} \]

   if -1.56e7 < y < 1.7000000000000001e35

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z\right) + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + \left(x + z\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y} + \left(x + z\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, x + z\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1}, y, x + z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y} + 1, y, x + z\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right)}, y, x + z\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z}, y, 1\right), y, x + z\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right)}, y, 1\right), y, x + z\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \color{blue}{\frac{-1}{2} \cdot z}\right), y, 1\right), y, x + z\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right), y, 1\right), y, \color{blue}{z + x}\right) \]

      12. lower-+.f64 94.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, \color{blue}{z + x}\right) \]

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot y, y, 1\right), y, z + x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, z + x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15600000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4600000000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+119}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4600000000.0) (+ x z) (if (<= y 2.95e+119) (+ (+ x y) z) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4600000000.0) {
		tmp = x + z;
	} else if (y <= 2.95e+119) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4600000000.0d0)) then
        tmp = x + z
    else if (y <= 2.95d+119) then
        tmp = (x + y) + z
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4600000000.0) {
		tmp = x + z;
	} else if (y <= 2.95e+119) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4600000000.0:
		tmp = x + z
	elif y <= 2.95e+119:
		tmp = (x + y) + z
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4600000000.0)
		tmp = Float64(x + z);
	elseif (y <= 2.95e+119)
		tmp = Float64(Float64(x + y) + z);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4600000000.0)
		tmp = x + z;
	elseif (y <= 2.95e+119)
		tmp = (x + y) + z;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4600000000.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 2.95e+119], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e9 or 2.95e119 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z + x} \]

      2. lower-+.f64 43.2

         \[\leadsto \color{blue}{z + x} \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{z + x} \]

   if -4.6e9 < y < 2.95e119

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} + z \]

      4. lower-+.f64 87.6

         \[\leadsto \color{blue}{\left(y + x\right)} + z \]

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\left(y + x\right) + z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4600000000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+119}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+47}:\\ \;\;\;\;z + y\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7e+47) (+ z y) (if (<= z 1.05e+60) (+ x y) (+ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e+47) {
		tmp = z + y;
	} else if (z <= 1.05e+60) {
		tmp = x + y;
	} else {
		tmp = z + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7d+47)) then
        tmp = z + y
    else if (z <= 1.05d+60) then
        tmp = x + y
    else
        tmp = z + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e+47) {
		tmp = z + y;
	} else if (z <= 1.05e+60) {
		tmp = x + y;
	} else {
		tmp = z + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7e+47:
		tmp = z + y
	elif z <= 1.05e+60:
		tmp = x + y
	else:
		tmp = z + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7e+47)
		tmp = Float64(z + y);
	elseif (z <= 1.05e+60)
		tmp = Float64(x + y);
	else
		tmp = Float64(z + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e+47)
		tmp = z + y;
	elseif (z <= 1.05e+60)
		tmp = x + y;
	else
		tmp = z + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7e+47], N[(z + y), $MachinePrecision], If[LessEqual[z, 1.05e+60], N[(x + y), $MachinePrecision], N[(z + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.00000000000000031e47 or 1.0500000000000001e60 < z

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} + z \]

      4. lower-+.f64 64.3

         \[\leadsto \color{blue}{\left(y + x\right)} + z \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\left(y + x\right) + z} \]

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.8%

      \[\leadsto z + \color{blue}{y} \]

   if -7.00000000000000031e47 < z < 1.0500000000000001e60

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} + z \]

      4. lower-+.f64 61.9

         \[\leadsto \color{blue}{\left(y + x\right)} + z \]

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\left(y + x\right) + z} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.9%

      \[\leadsto y + \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+47}:\\ \;\;\;\;z + y\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x z))

C

double code(double x, double y, double z) {
	return x + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + z
end function

Java

public static double code(double x, double y, double z) {
	return x + z;
}

Python

def code(x, y, z):
	return x + z

Julia

function code(x, y, z)
	return Float64(x + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + z;
end

Wolfram

code[x_, y_, z_] := N[(x + z), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + z} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{z + x} \]

   2. lower-+.f64 66.1

      \[\leadsto \color{blue}{z + x} \]

5. Applied rewrites 66.1%

   \[\leadsto \color{blue}{z + x} \]

6. Final simplification 66.1%

   \[\leadsto x + z \]
7. Add Preprocessing

Alternative 12: 38.4% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x y))

C

double code(double x, double y, double z) {
	return x + y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + y
end function

Java

public static double code(double x, double y, double z) {
	return x + y;
}

Python

def code(x, y, z):
	return x + y

Julia

function code(x, y, z)
	return Float64(x + y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + y;
end

Wolfram

code[x_, y_, z_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + \left(y + z\right)} \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(x + y\right) + z} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + y\right) + z} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(y + x\right)} + z \]

   4. lower-+.f64 62.9

      \[\leadsto \color{blue}{\left(y + x\right)} + z \]

5. Applied rewrites 62.9%

   \[\leadsto \color{blue}{\left(y + x\right) + z} \]

6. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{y} \]
7. Step-by-step derivation

8. Applied rewrites 37.4%

   \[\leadsto y + \color{blue}{x} \]

9. Final simplification 37.4%

   \[\leadsto x + y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))